SAT-based versus CSP-based Constraint Weighting for Satisfiability
Duc Nghia Pham and John Thornton and Abdul Sattar and Abdelraouf Ishtaiwi
Institute for Integrated and Intelligent System
Griffith University, Queensland, Australia
email:{d.n.pham, j.thornton, a.sattar, a.ishtaiwi}@griffith.edu.au
Abstract
Recent research has focused on bridging the gap between the satisfiability (SAT) and constraint satisfaction
problem (CSP) formalisms. One approach has been to
develop a many-valued SAT formula (MV-SAT) as an
intermediate paradigm between SAT and CSP, and then
to translate existing highly efficient SAT solvers to the
MV-SAT domain. Experimental results have shown this
approach can achieve significant improvements in performance compared with the traditional SAT and CSP
approaches.
In this paper, we follow a different route, developing
SAT solvers that can automatically recognise CSP structure hidden in SAT encodings. This allows us to look
more closely at how constraint weighting can be implemented in the SAT and CSP domains. Our experimental results show that a SAT-based approach to handle weights, together with CSP-based approach to variable instantiation, is superior to other combinations of
SAT and CSP-based approaches. A further experiment
on the round robin scheduling problem indicates that
this many-valued constraint weighting approach outperforms other state-of-the-art solvers.
Introduction
Constraint satisfaction is an intuitively simple but expressively powerful concept. Over the last three decades, it has
emerged as a major research field within artificial intelligence and computer science. Much of this research has focused on two areas: firstly the modelling of complex problems as constraint satisfaction problems (CSPs), e.g. in planning and scheduling, scene analysis, combinatorial problems, etc., and secondly the development of search techniques for efficiently solving such problems. While modelling hard real-world problems as CSPs remains an interesting research challenge (e.g., in bioinformatics), it has recently been recognised that many CSPs can be reformulated
as satisfiability (SAT) problems and solved more efficiently
using modern SAT solvers (Hoos 1999b; Walsh 2000). Examples of such problem domains include planning (Kautz
& Selman 1996), scheduling (Béjar & Manyà 2000), hardware verification (Velev & Bryant 2003) and combinatorial
problems (Kautz et al. 2001).
c 2005, American Association for Artificial IntelliCopyright °
gence (www.aaai.org). All rights reserved.
However, the encoding of CSPs as SAT problems raises
further issues: firstly, many real world problems are more
naturally formulated using non-Boolean variables, i.e. variables that can have more than two values. This means a SAT
encoding can result in exponential increases in problem size.
Secondly, the structure of a CSP, when reformulated as SAT,
is usually flattened out and remains hidden in the encoding.
This loss of structure can have negative impacts on solver
performance. These concerns have motivated the investigation of hybrid models of SAT and CSP representations. For
example, the automated theorem proving community has
studied many-valued clausal formulas that represent conjunctive normal forms (CNF) of many-valued domain variables (Ansótegui et al. 2003). This representation enables
SAT solvers to exploit the highly compact and structured nature of the CSP variables. More recent studies have looked
at bridging the gap between SAT and CSP formalisms, and
have produced a new generation of powerful many valued (MV-SAT) solvers, such as Mv-Satz (Ansótegui et al.
2003), a many-valued variant of Chaff (Ansótegui, Larrubia,
& Manyà 2003), NB-WalkSAT (Frisch & Peugniez 2001)
and Regular-SAT (Béjar et al. 2001). These methods can
solve hard combinatorial problems, such as all interval series, round robin scheduling and quasigroup completion at a
level unmatched by existing standard SAT and CSP solvers.
On the other hand, stochastic local search (SLS) SAT
solvers have been significantly improved in recent years.
Within this group, dynamic local search (DLS) techniques
(Hoos & Stützle 2005) have proved the most promising,
especially for larger and more difficult problems. The underlying idea of a DLS approach is to dynamically adjust
the weights of false clauses during the search, thereby escaping local minima and moving quickly towards a solution. The superiority of such DLS algorithms over nonweighting SLS solvers has been demonstrated on a wide
range of SAT benchmarks (Hutter, Tompkins, & Hoos 2002;
Thornton et al. 2004), but as yet DLS has not been explicitly applied to the many valued SAT domain. From this,
two questions arise: firstly, will the superior performance of
DLS algorithms still be maintained in the MV-SAT domain?
And secondly, what is the most suitable way for an MV-SAT
DLS solver to handle clause weights?
In this paper, we aim to answer these questions through
an investigation of the behaviour and performance of PAWS
AAAI-05 / 455
(Thornton et al. 2004) and SAPS (Hutter, Tompkins, &
Hoos 2002) in the MV-SAT domain. We chose PAWS and
SAPS because they represent the state-of-the-art for DLS
SAT solving using additive and multiplicative weighting respectively. We firstly extend the method of Ansótegui, Larrubia, & Manyà (2003) to make these SAT solvers automatically recognise the underlying structure of CSP variables and constraints hidden inside SAT formulas. Based
on this recovered knowledge, we modified the SAT versions
of PAWS and SAPS in small steps until they became CSPlike DLS algorithms. This approach allows us to look more
closely at how constraint weighting should be handled in the
MV-SAT domain. Our experimental results show that a SATbased approach to handle weights, together with CSP-based
approach to variable selection, is superior to other combinations of SAT and CSP-based approaches. Our empirical results on the round robin scheduling problem also indicate that the proposed many-valued constraint weighting
approach outperforms other state-of-the-art SAT solvers.
The remainder of the paper is structured as follows: the
next section reviews and discusses how modern DLS algorithms handle weights in the SAT domain. We then describe how to make SAT solvers automatically recognise
hidden CSP structure. Further, we describe how constraint
weighting should be handled in the many-valued domain by
comparing the effects of SAT and CSP-based approaches to
weight adjustment in combination with the SAT and CSPbased variable instantiation methods. In the next section we
discuss our experimental study, describing the experimental
design and analysing the performance results. Finally, we
conclude the paper with some remarks on related and future
work.
Dynamic Local Search for SAT
Like other SLS techniques, DLS algorithms start with a random truth assignment for each Boolean variable in a given
formula, and iteratively flip single literals that minimise the
objective function until a solution is found or a local minimum is encountered. The basics of DLS are sketched in
algorithm 1. The main idea is that weights are typically associated with the clauses of a given formula and the sum of
weights of unsatisfied clauses is used as the objective function to select the next move. During the search, DLS solvers
dynamically adjust the clause weights and hence modify the
search landscape to effectively avoid or escape from local
minima.
Since the introduction of the Breakout heuristic (Morris 1993), DLS algorithms have evolved into several variants which differ in which clause weights should be updated
(all clauses or only unsatisfied clauses), how these weights
should be adjusted (additively or multiplicatively), and when
weight updates should take place (deterministically or probabilistically) (Hoos & Stützle 2005). Despite these differences, the underlying strategy to escape from local minima is based on two mechanisms: increasing and reducing
weights.1
1
Also known as scaling and smoothing in multiplicative weighting DLS algorithms.
Algorithm 1 DLS(F)
generate a random starting point;
set the weight wi of each clause ci to 1;
while solution is not found and not timeout do
find a list L of variables that minimise Σwi if flipped;
if such list L exists then
randomly flip a variable in L;
else
update the weight of each false clause;
end if
end while
When a DLS search procedure encounters a local minimum, weights are increased on the currently unsatisfied
clauses, changing the surface of the search landscape. As a
result, the search is forced to move to a new neighbour that
can satisfy at least one of the currently unsatisfied clauses,
thus escaping from the original local minimum. However,
this increasing mechanism has side-effects: as it locally
modifies the search landscape to escape the current local
minimum, it possibly gives rise to other new local minima,
which may in some cases be even harder to avoid or escape
(Morris 1993). Therefore, the reducing mechanism was introduced to counter these side-effects. After a certain period
of time, the weights of chosen clauses are reduced in order
to make the search forget the high costs of violating clauses
which are no longer helpful.
CSP-based Dynamic Local Search for SAT
In this section we describe the procedures used to integrate
the CSP-based variable instantiation heuristic and the CSPbased weighting mechanism into SAPS and PAWS:
CSP Variable Extraction
In (2003), Ansótegui, Larrubia, & Manyà hypothesised that
SAT solvers could take advantage of the domain structure
of CSP variables. They embedded a detection mechanism
into Chaff to automatically identify the set of Boolean variables that model the same CSP variable. Their experiments showed that the performance of Chaff is significantly
boosted by the integration of a CSP-based variable instantiation heuristic.
In a direct SAT encoding, a CSP variable Xi with a domain Di = {1, 2, . . . , m} is encoded using a set of m
Boolean variables Bi = {xsi |s ∈ [1..m]} so that the truth
value of xsi represents the assignment of value s to Xi . The
following clauses are also added to the SAT formula to ensure the exclusive-or relationship of CSP variable-value assignments, i.e. the attribute that Xi can be instantiated with
exactly one value of Di :
• One at-least-one (ALO) clause, x1i ∨ . . . ∨ xm
i , to ensure
that at least one domain value is assigned to Xi ; and
• A set of at-most-one (AMO) clauses, ¬xsi ∨ ¬xti where
1 ≤ s ≤ t ≤ m, to ensure that at most one domain value
can be assigned to Xi .
AAAI-05 / 456
By detecting these two types of clauses in a given SAT
formula, we can recognise the sets of Boolean variables that
model the underlying CSP variables. It is worth noting that
several studies on SAT encodings of CSPs have reported an
improvement of performance when applying SLS solvers to
a direct SAT encoding without the AMO clauses (known as
a multivalued encoding (Prestwich 2004)).2 A CSP solution
can then easily be extracted from a SAT solution by taking
any SAT-assigned value for each CSP variable.
CSP-based Variable Instantiation Scheme
We firstly integrated the CSP variable extraction mechanism into SAPS and PAWS to automatically extract the underlying CSP variable structures from the SAT encodings
(with or without the presence of AMO clauses). We then
modified the variable instantiation mechanism, which controls how a Boolean variable is selected and flipped, so
that the exclusive-or relationship was built into the operation of the algorithms. This is related to the work on
NB-WalkSAT (Frisch & Peugniez 2001) and Regular-SAT
(Béjar et al. 2001), that similarly enforces the unique instantiation of variables with non-binary domains within an
MV-SAT framework.
The new CSP-based variable instantiation heuristic ensures that for each underlying CSP variable Xi there is exactly one true Boolean variable xti at any one time during the
search, while all related Boolean variables xsi ∈ Bi − {xti }
remain false. Each time the search looks for a Boolean variable to flip, all false variables xsi involved in false clauses,
and all associated variables xui ∈ Bi − {xti , xsi } are considered for selection. This differs from the original SAT-based
variable instantiation heuristic which ignores these associated variables. When a false variable xsi is flipped, the previously associated true variable xti is set to false in one operation. Hence all ALO and AMO clauses become redundant
and can be removed from the problem.
CSP-based Weight Updating Scheme
In a binary CSP, a constraint Cij between two CSP variables
Xi and Xj defines a set of paired domain values (s, t) that
Xi and Xj cannot take simultaneously. In the direct SAT
encoding of a binary CSP, such a constraint Cij is encoded
using a set of conflict (CON) clauses. A CON clause cst
ij , of
the form ¬xsi ∨ ¬xtj , ensures that if Xi is instantiated with
value s then Xj cannot be instantiated with value t, or vice
versa.
In a binary CSP, weight is generally added to unsatisfied constraints, whereas in a direct SAT encoding weight is
added to particular pairs of values that violate the underlying
CSP constraint. Hence, in a CSP approach, all possible instantiations that violate a constraint are penalised, whereas
in a SAT approach, only the current violating instantiation
is penalised. In separate research, a similar fine-grained
weighting scheme was employed in the GENET artificial
neural network CSP solver (Choi, Lee, & Stuckey 2000).
It is still an open research question as to which weighting
approach is superior for any given problem class.
In order to answer this question, we integrated a CSP constraint extraction mechanism into SAPS and PAWS to automatically recognise the set of CON clauses that represents
the underlying CSP constraint. We then modified the weight
updating mechanism in SAPS and PAWS so that, in both the
weight increasing and reducing phases, either the weight of a
particular unsatisfied clause cst
ij is updated or the weights of
all clauses that represent an unsatisfied CSP constraint Cij
are updated. With the new architecture, we are able to compare different variants of weighting approaches based on the
combinations between SAT-based and CSP-based variable
instantiation and weighting mechanisms.
Experimental Results and Discussion
Based on this new architecture, we implemented four variants of SAPS and four variants of PAWS, using combinations of the SAT and CSP-based heuristics described above:
• SVI+SWM (ss): SAT-based variable instantiation and
SAT-based weighting mechanism (the original DLS SAT
implementation);
• SVI+CWM (sc): SAT-based variable instantiation and
CSP-based weighting mechanism;
• CVI+SWM (cs): CSP-based variable instantiation and
SAT-based weighting mechanism;
• CVI+CWM (cc): CSP-based variable instantiation and
CSP-based weighting mechanism (a total CSP-like DLS
variant for SAT);
As one of the aims of this paper is to evaluate the impact
of SAT and CSP-based weighting heuristics on the performance of DLS SAT solvers, but not to compare additive with
multiplicative weighting, we selected the reactive version of
SAPS (RSAPS) and used the default values for α (1.3) and ρ
(0.8) to avoid the time consuming parameter tuning required
for SAPS.3 Note also that Hutter, Tompkins, & Hoos (2002)
reported that the performance of RSAPS is comparable to
SAPS on a wide range of benchmark problems.
Problem Set
We selected all-interval-series, graph colouring, uniform
random binary CSPs and round robin problems as the benchmarks to evaluate the performance of the eight DLS SAT
solver variants.
We obtained the graph colouring (flat100 and flat200) and
random binary CSPs from the authors of SAPS and PAWS.
To further evaluate the scalability of the DLS variants, we
generated 10 instances of the flat graph colouring problem
using Culberson’s generator with 450 vertices, 3 colours
and an edge density of 0.018.4 This is the same generator
used for the flat100 and flat200 problems. We then translated these 10 instances into direct SAT encodings and ran
3
2
This case was not considered in (Ansótegui, Larrubia, &
Manyà 2003).
Three out of four parameters of SAPS require fine tuning to
achieve optimal performance of SAPS (Thornton et al. 2004).
4
http://web.cs.ualberta.ca/∼joe/Coloring/
AAAI-05 / 457
7
7
PAWS Scalability
x 10
6
SS
SC
CS
CC
5
4
3
2
1
0
100−m 100−h 200−m 200−h 450−m 450−h
Problem (by difficulty)
0
100−m 100−h 200−m 200−h 450−m 450−h
Problem (by difficulty)
Figure 1: Scalability of PAWS and RSAPS variants on flat
problems. The problems are ordered by hardness. The N m and N -h stand for the median and hard instances of the
flatN series.
PAWS Run−Time Distribution
SAPS Run−Time Distribution
100
SS
SC
CS
CC
80
60
40
20
0
0
200
400 600 800
Time (in seconds)
1000 1200
Percent successful runs
Table 1 presents the results of PAWS and RSAPS variants
on the structured CSP benchmarks, consisting of the allinterval-series and graph colouring problems. These results
show the CVI+SWM combination consistently dominates
the other combinations, both within the PAWS and RSAPS
results and across problem classes.
By grouping the four variants into a CVI-based and an
SVI-based set, we can consider the impact of the different
weighting mechanisms on the performance of RSAPS and
PAWS, and ignore the effects of the variable instantiation
heuristics. Such an analysis shows that in both sets the performance of the SWM variants are up to order of magnitude better than the CWM variants both in terms of execution time and local search cost (the number of flips performed to find a solution). Hence, SWM emerges as the best
weighting approach for all-interval-series and graph colouring problems.
We used the same approach to evaluate the impact of the
different variable instantiation heuristics on the performance
of PAWS and RSAPS. In contrast to the weighting methods,
the CSP-based variable instantiation heuristic emerges as
significantly better than the SAT-based heuristic. For PAWS,
the CSP-based variants performed about 10 times better than
the SAT-based variants on the flat450 series in terms of execution time and around 30 times better in terms of local
search cost. For RSAPS, the relative dominance of the CSPbased variants was even more marked.
Figure 1 graphs the mean local search costs of the RSAPS
variants and PAWS variants on the flat graph colouring series. These two graphs clearly indicate that the performance
of the CVI-based variants scale better than the SVI-based
variants as the problems become harder and larger.
In the random binary CSP domain, the CVI+SWM variant
is generally still the better of the four variants, although for
PAWS the performance of the CVI+CWM variant is close
to the CVI+SWM variant and even slightly better in terms
of the local search cost. However, the PAWS’ CVI+SWM
variant is still better in terms of execution time, as shown in
the run-time distributions (RTDs) (Hoos 1999a) of Figure 2.
In contrast to the results for the structured CSPs, Table 2 shows a reversal in the relative performance between
Percent successful runs
100
http://www.satlib.org
3
1
Structured and Random Binary CSPs
5
SS
SC
CS
CC
4
2
SAPS Scalability
x 10
5
Flips
6
Flips
RSAPS to determine the median and hardest problem instances. Other problems used in this study are available at
SATLIB.5
The results of the PAWS and SAPS variants on these
benchmark problems are shown in Tables 1, 2 and 3. Each
variant was run 1, 000 times on each benchmark problem
except the 50v15d40c problems (100 runs) and the round
robin scheduling for 16, 18 and 20 teams (with 100, 50 and
10 runs, respectively). All the time results are in seconds unless otherwise stated, and all experiments were performed on
a Sun supercomputer with 8 × Sun Fire V880 servers, each
with 8 × UltraSPARC-III 900MHz CPU and 8GB memory
per node.
SS
SC
CS
CC
80
60
40
20
0
0
200
400 600 800
Time (in seconds)
1000 1200
Figure 2: RTDs of PAWS and RSAPS variants on the
50v15d40c-hard instance.
SAT-weighting and CSP-weighting when using the SATvariable instantiation. Here, RSAPS prefers CSP-weighting
on all the random binary problems, and PAWS prefers CSPweighting on the harder, larger problems. However, as the
SAT-variable instantiation is not competitive, this result is of
minor interest.
Round Robin Scheduling Problems
The results in the first experiment indicate that a combined
application of CSP-based variable instantiation and a SATbased weighting mechanism can significantly boost the performance of DLS SAT solvers. To further clarify the efficiency of this approach in comparison with other stateof-the-art SAT solvers, we ran a second experiment on
the round robin scheduling problem. In recent years, this
sport tournament problem has become an important benchmark for the combinatorial search community (Gomes et al.
1998). In 2000, Béjar & Manyà translated this problem into
the SAT domain and was able to find a solution for 20 teams
in an average of 12.73 hours using the R-Novelty algorithm.
Before this work, the 20 team instance was unsolvable by
either combinatorial, SAT or MV-SAT approach.
For this experiment, we ran the CVI+SWM variants of
RSAPS and PAWS, called MV-RSAPS and MV-PAWS, on
seven round robin problem instances from 8 teams to 20
teams (these problems were originally used in (Béjar &
Manyà 2000)6 ). Table 3 shows the results of MV-PAWS
and MV-RSAPS in comparison with the original PAWS
and RSAPS and two of the best complete SAT solvers,
6
AAAI-05 / 458
http://web.udl.es/usuaris/d4372149/
PAWS
Problem
RSAPS
PAWS
%
Time
Flips
%
Time
Flips
Time
Flips
%
Time
SVI+CWM
100%
0.14
37, 704
100%
0.14
34, 842
v=30
Flips
SVI+CWM
100%
0.06
9, 490
100%
0.14
23, 649
SVI+SWM
100%
0.08
24, 684
100%
0.07
20, 393
CVI+CWM
100%
0.05
5, 316
100%
0.04
4, 630
d=10
SVI+SWM
100%
0.04
6, 845
100%
0.21
43, 459
c=80
CVI+CWM
100%
0.02
1, 522
100%
0.03
CVI+SWM
100%
0.04
5, 040
100%
0.03
3, 063
2, 379
median
CVI+SWM
100%
0.02
1, 584
100%
0.03
SVI+CWM
100%
2.34
435, 026
100%
1.95
2, 216
394, 003
v=30
SVI+CWM
100%
0.08
14, 160
100%
0.20
38, 798
SVI+SWM
100%
1.17
250, 188
100%
CVI+CWM
100%
0.37
34, 830
100%
0.67
157, 082
d=10
SVI+SWM
100%
0.07
10, 298
100%
0.41
77, 230
0.36
27, 989
c=80
CVI+CWM
100%
0.03
2, 592
100%
0.05
CVI+SWM
100%
0.35
33, 000
4, 116
100%
0.16
14, 794
hard
CVI+SWM
100%
0.03
2, 447
100%
0.04
flat100- SVI+CWM
100%
0.01
2, 994
9, 170
100%
0.02
8, 663
v=30
SVI+CWM
100%
0.14
23, 813
100%
0.27
45, 249
median
SVI+SWM
100%
CVI+CWM
100%
0.01
8, 348
100%
0.01
7, 677
d=10
SVI+SWM
100%
0.07
14, 998
100%
0.40
84, 141
0.01
1, 644
100%
0.01
2, 134
c=40
CVI+CWM
100%
0.05
3, 882
100%
0.09
CVI+SWM
6, 844
100%
0.00
1, 606
100%
0.00
1, 890
median
CVI+SWM
100%
0.04
3, 811
100%
0.05
5, 094
flat100- SVI+CWM
100%
0.05
35, 614
100%
0.05
30, 025
v=30
SVI+CWM
100%
0.14
23, 171
100%
0.31
50, 710
hard
SVI+SWM
100%
0.05
37, 311
100%
0.06
30, 063
d=10
SVI+SWM
100%
0.11
23, 266
100%
0.48
98, 759
CVI+CWM
100%
0.02
7, 894
100%
0.02
8, 229
c=40
CVI+CWM
100%
0.05
4, 176
100%
0.09
6, 723
CVI+SWM
100%
0.02
7, 736
100%
0.02
8, 169
hard
CVI+SWM
100%
0.04
3, 729
100%
0.06
5, 063
flat200- SVI+CWM
100%
0.29
181, 566
100%
0.64
361, 786
v=50
SVI+CWM
100%
1.49
134, 128
99.8%
76.04
5, 631, 474
median
SVI+SWM
100%
0.19
162, 398
100%
0.49
296, 697
d=10
SVI+SWM
100%
1.37
132, 363
99.1%
CVI+CWM
100%
0.12
31, 786
100%
0.24
60, 235
c=80
CVI+CWM
100%
0.62
25, 575
100%
2.49
87, 653
CVI+SWM
100%
0.08
25, 200
100%
0.20
48, 278
median
CVI+SWM
100%
0.60
29, 590
100%
1.47
59, 583
flat200- SVI+CWM
100%
10.03
6, 819, 680
100%
9.05
4, 807, 225
v=50
SVI+CWM
100%
1.90
159, 213
99.8%
92.65
6, 892, 970
hard
SVI+SWM
100%
4.39
3, 153, 438
100%
6.06
4, 148, 664
d=10
SVI+SWM
100%
1.62
155, 124
98.9%
CVI+CWM
100%
1.73
573, 207
100%
2.74
676, 973
c=80
CVI+CWM
100%
0.74
30, 196
100%
3.17
110, 510
CVI+SWM
100%
1.18
426, 136
100%
1.62
461, 815
hard
CVI+SWM
100%
0.67
33, 221
100%
1.71
68, 682
flat450- SVI+CWM
100%
54.05 15, 407, 113
100%
99.42 28, 004, 978
v=50
SVI+CWM
100% 169.42 11, 878, 897
54%
729.61 58, 600, 553
median
SVI+SWM
100%
24.25
7, 902, 070
100%
56.56 15, 966, 332
d=10
SVI+SWM
100% 192.58 17, 196, 220
44%
889.19 78, 820, 237
CVI+CWM
100%
2.72
341, 017
100%
33.00
3, 135, 384
c=40
CVI+CWM
100%
47.12
1, 593, 711
99%
84.20
2, 722, 482
CVI+SWM
100%
2.22
317, 243
100%
12.11
1, 203, 609
median
CVI+SWM
100%
35.04
1, 542, 803
100%
38.32
1, 442, 271
100% 226.70 15, 813, 749
37%
ais12
Problem
Variant
RSAPS
%
ais10
Variant
130.43 11, 586, 967
152.19 13, 704, 204
flat450- SVI+CWM
100% 190.24 55, 363, 281
93.2% 192.16 54, 140, 275
v=50
SVI+CWM
hard
SVI+SWM
100% 107.65 34, 479, 371
99.4% 111.74 31, 867, 483
d=10
SVI+SWM
CVI+CWM
100%
13.29
1, 643, 087
100%
73.29
6, 838, 029
c=40
CVI+CWM
100%
72.99
2, 522, 204
93%
225.51
8, 200, 102
CVI+SWM
100%
12.36
1, 672, 053
100%
36.78
3, 574, 930
hard
CVI+SWM
100%
58.75
2, 664, 852
100%
90.90
3, 455, 520
g125.17 SVI+CWM
100%
22.56
1, 348, 325
0% 600.00
n/a
SVI+SWM
100%
9.82
744, 124
0% 600.00
n/a
CVI+CWM
100%
5.24
166, 449
100% 371.78
8, 829, 445
100%
CVI+SWM
100%
4.53
159, 275
60.85
1, 718, 548
g250.29 SVI+CWM
100%
69.23
579, 887
0% 600.00
n/a
SVI+SWM
100%
28.71
359, 073
0% 600.00
n/a
CVI+CWM
100%
13.86
79, 135
0% 600.00
n/a
CVI+SWM
100%
12.36
75, 718
94% 215.40
1, 174, 523
Table 1: Structured CSP problems
zChaff.2004.11.15 and Satz215. We also include the results
of R-Novelty reported in (Béjar & Manyà 2000), in which
the top numbers are the original results and the bottom numbers are the approximated results if run on our test machine.
These results show MV-PAWS is a clear winner, particularly
on the larger problems where it can find a solution for the
20 team instance within one hour, whereas R-Novelty takes
more than 3 hours.
Conclusions and Future Work
We have integrated a structure extraction mechanism into
SAPS and PAWS that automatically recovers the underlying
CSP variable and constraint structure hidden in SAT formu-
99% 214.15 19, 248, 169
954.80 75, 526, 227
28% 1047.34 91, 501, 564
Table 2: Random binary CSP problems
las. Based on this new architecture, we are able to explore
and evaluate the impact of different SAT and CSP-based
heuristics for instantiating variables and handling weights on
the performance of DLS SAT solvers. Our experimental results show that a SAT-based weighting mechanism together
with a CSP-based variable instantiation is the most suitable DLS approach to solve direct SAT-encodings of CSPs.
A further comparison on the round robin scheduling problem shows that this approach significantly outperforms other
methods. In particular, using the MV-PAWS variant with the
CVI and SWM heuristics, we can find solutions for the 20
team round robin scheduling problem within one hour, while
the previous state-of-the-art approach required several hours
to solve the same problem.
In 2002, Ostrowski et al. produced a pioneering paper
on recognising the variable dependencies in SAT formulas through logic gates of the form {⇔, ∧, ∨}. Within this
framework, the unique instantiation attribute of CSP variables discussed in this paper can be seen as an exclusive-or
gate. In future work, we intend to further integrate the use
of gates into SLS SAT solvers, especially DLS techniques.
AAAI-05 / 459
Succ.
Problem
n=8
Variant
MV-PAWS
MV-RSAPS
PAWS
RSAPS
n = 10 MV-PAWS
MV-RSAPS
PAWS
RSAPS
n = 12 MV-PAWS
MV-RSAPS
PAWS
RSAPS
n = 14 MV-PAWS
MV-RSAPS
PAWS
RSAPS
n = 16 MV-PAWS
MV-RSAPS
PAWS
RSAPS
n = 18 MV-PAWS
MV-RSAPS
PAWS
RSAPS
n = 20 MV-PAWS
MV-RSAPS
PAWS
RSAPS
Time
Flips
Param
%
mean
mean
Time
Time
Satz
zChaf f
R-Novelty
35
100%
0.01
610
1.85
n/a
100%
0.01
627
0.14
3
100%
0.02
2, 879
n/a
100%
0.16
18, 428
12
100%
0.16
5, 057
17.24
n/a
100%
0.35
9, 694
3190.37
3
100%
0.39
24, 177
n/a
100%
70.85
4, 236, 607
n/a
n/a
11
100%
2.29
36, 912
115.27
16.20
n/a
100%
7.41
90, 947
> 24hrs
4.50
3
100%
9.01
258, 507
n/a
11%
3419.69 90, 818, 167
8
100%
26.66
242, 734
> 24hrs
104.40
n/a
89.8%
253.71
1, 528, 434
> 24hrs
29.00
3
58%
423.70
5, 481, 201
0% > 24hrs
n/a
209.71
1, 352, 252
> 24hrs
1008.00
60% 16535.46 74, 040, 590
> 24hrs
280.00
n/a
6
n/a
3
n/a
100%
0% > 24hrs
n/a
0% > 24hrs
n/a
1652.95
6, 985, 404
> 24hrs
7128.00
n/a
0% > 24hrs
n/a
> 24hrs
1980.00
3
0% > 24hrs
n/a
5
n/a
100%
0% > 24hrs
n/a
3177.17
9, 079, 655
> 24hrs 45828.00
n/a
0% > 24hrs
n/a
> 24hrs 12730.00
3
0% > 24hrs
n/a
n/a
0% > 24hrs
n/a
4
100%
Table 3: Round robin results
Another interesting direction is to extend this work to different SAT encodings of CSPs. Gent (2002) looked at the
SAT support encoding, which allows complete SAT solvers
to maintain the arc consistency of CSP problems by using
unit propagation. He reported that local search techniques
such as WalkSAT can also benefit from this encoding. It
would therefore be interesting to investigate the behaviour
and performance of SAT and CSP-based variants of DLS
SAT solvers using the support encoding.
References
Ansótegui, C.; Larrubia, J.; Li, C. M.; and Manyà, F. 2003.
Mv-Satz: A SAT solver for many-valued clausal forms. In
Proceedings of JIM-2003.
Ansótegui, C.; Larrubia, J.; and Manyà, F. 2003. Boosting
Chaff’s performance by incorporating CSP heuristics. In
Proceedings of CP-2003, 96–107.
Béjar, R., and Manyà, F. 2000. Solving the round
robin problem using propositional logic. In Proceedings
of AAAI-2000, 262–266.
Béjar, R.; Cabiscol, A.; Fernández, C.; Manyà, F.; and
Gomes, C. P. 2001. Capturing structure with satisfiability. In Proceedings of CP-2001, 137–152.
Choi, K.; Lee, J. H.-M.; and Stuckey, P. J. 2000. A La-
grangian reconstruction of GENET. Artificial Intelligence
123:201–215.
Frisch, A., and Peugniez, T. 2001. Solving non-Boolean
satisfiability problems with stochastic local search. In Proceedings of IJCAI-2001.
Gent, I. P. 2002. Arc consistency in SAT. In Proceedings
of ECAI-02, 121–125.
Gomes, C.; Selman, B.; McAloon, K.; and Tretkoff, C.
1998. Randomization in backtrack search: Exploiting
heavy-tailed profiles for solving hard scheduling problems.
In Proceedings of AIPS-1998.
Hoos, H. H., and Stützle, T. 2005. Stochastic Local Search:
Foundations and Applications. San Francisco, CA: Morgan
Kaufmann.
Hoos, H. H. 1999a. On the run-time behaviour of stochastic local search algorithms for SAT. In Proceedings of
AAAI-1999, 661–666.
Hoos, H. H. 1999b. SAT-encodings, search space structure,
and local search performance. In Proceedings of IJCAI1999, 296–302.
Hutter, F.; Tompkins, D. A. D.; and Hoos, H. H. 2002.
Scaling and probabilistic smoothing: Efficient dynamic local search for SAT. In Proceedings of CP-2002, 233–248.
Kautz, H., and Selman, B. 1996. Pushing the envelope:
Planning, propositional logic, and stochastic search. In
Proceedings of AAAI-1996, 1194–1201.
Kautz, H.; Ruan, Y.; Achlioptas, D.; Gomes, C.; Selman,
B.; and Stickel, M. 2001. Balance and filtering in structured satisfiable problems. In Proceedings of IJCAI-2001,
351–358.
Morris, P. 1993. The Breakout method for escaping from
local minima. In Proceedings of AAAI-1993, 40–45.
Ostrowski, R.; Grégoire, E.; Mazure, B.; and Saı̈s, L. 2002.
Recovering and exploiting strutural knowledge from CNF
formulas. In Proceedings of CP-2002, 185–199.
Prestwich, S. 2004. Full dynamic substitutability by SAT
encoding. In Proceedings of CP-2004, 512–526.
Thornton, J.; Pham, D. N.; Bain, S.; and Ferreira Jr., V.
2004. Additive versus multiplicative clause weighting for
SAT. In Proceedings of AAAI-2004, 191–196.
Velev, M. N., and Bryant, R. E. 2003. Effective use of
Boolean satisfiability procedures in the formal verification
of superscalar and VLIW microprocessors. Journal of Symbolic Computation 35(2):73–106.
Walsh, T. 2000. SAT v CSP. In Proceedings of CP-2000,
441–456.
AAAI-05 / 460